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A stochastic fluid-structure interaction problem with the Navier slip boundary condition

Krutika Tawri

Abstract

We prove the existence of martingale solutions to a stochastic fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the Navier-Stokes equations, through a deformable elastic tube modeled by shell/membrane equations. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic coupling conditions at the fluid-structure interface. This article considers the case where the structure can have unrestricted displacement and explores the Navier-slip boundary condition imposed at the fluid-structure interface, displacement of which is not known a priori and is itself a part of the solution. The proof takes a constructive approach based on a Lie splitting scheme. The geometric nonlinearity stemming from the nonlinear coupling, the possibility of random fluid domain degeneracy, the potential jumps in the tangential components of the fluid and structure velocities at the moving interface and the low regularity of the structure velocity require the development of new techniques that lead to the existence of martingale solutions.

A stochastic fluid-structure interaction problem with the Navier slip boundary condition

Abstract

We prove the existence of martingale solutions to a stochastic fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the Navier-Stokes equations, through a deformable elastic tube modeled by shell/membrane equations. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic coupling conditions at the fluid-structure interface. This article considers the case where the structure can have unrestricted displacement and explores the Navier-slip boundary condition imposed at the fluid-structure interface, displacement of which is not known a priori and is itself a part of the solution. The proof takes a constructive approach based on a Lie splitting scheme. The geometric nonlinearity stemming from the nonlinear coupling, the possibility of random fluid domain degeneracy, the potential jumps in the tangential components of the fluid and structure velocities at the moving interface and the low regularity of the structure velocity require the development of new techniques that lead to the existence of martingale solutions.
Paper Structure (13 sections, 21 theorems, 164 equations, 1 figure)

This paper contains 13 sections, 21 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setup
  3. The deterministic model and a weak formulation
  4. Stochastic framework on fixed domain
  5. ALE mappings
  6. Definition of martingale solutions
  7. Operator splitting scheme
  8. Definition of the splitting scheme
  9. Approximate solutions
  10. Passing $N\rightarrow\infty$
  11. Tightness results
  12. Almost sure convergence
  13. Passing to the limit $\varepsilon\to 0$

Key Result

Lemma 3.1

Consider $v\in H^{2}_0(\Gamma)$ and $w \in H^2(\mathcal{O})$ that solve, Then,

Figures (1)

  • Figure 1: A realization of the fluid domain

Theorems & Definitions (36)

  • Definition 1
  • Lemma 3.1
  • proof
  • Remark 1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • proof
  • ...and 26 more